Question bank AMC10 2002

The ratio $$
\dfrac{10^{2000}+10^{2002}}{10^{2001}+10^{2001}}
$$ is closest to which of the following numbers?

For the nonzero numbers $a,$ $b,$ and $c,$ define $$
(a,b,c)=\dfrac{a}{b}+\dfrac{b}{c}+\dfrac{c}{a}.
$$ Find $(2,12,9).$

According to the standard convention for exponentiation, $$
2^{2^{2^{2}}}=2^{\left(2^{\left(2^{2}\right)}\right)}=2^{16}=65{,}536.
$$ If the order in which the exponentiations are performed is changed, how many *other* values are possible?

For how many positive integers $m$ does there exist at least one positive integer $n$ such that $m\cdot n\le m+n?$

Each of the small circles in the figure has radius one. The innermost circle is tangent to the six circles that surround it, and each of those circles is tangent to the large circle and to its small-circle neighbors. Find the area of the shaded region.

Cindy was asked by her teacher to subtract $3$ from a certain number and then divide the result by $9.$ Instead, she subtracted $9$ and then divided the result by $3,$ giving an answer of $43.$ What would her answer have been had she worked the problem correctly?

If an arc of $45^\circ$ on circle $A$ has the same length as an arc of $30^\circ$ on circle $B,$ then the ratio of the area of circle $A$ to the area of circle $B$ is

Betsy designed a flag using blue triangles, small white squares, and a red center square, as shown. Let $B$ be the total area of the blue triangles, $W$ the total area of the white squares, and $R$ the area of the red square. Which of the following is correct?

Suppose $A,$ $B,$ and $C$ are three numbers for which $1001C-2002A=4004$ and $1001B+3003A=5005.$ The average of the three numbers $A,$ $B,$ and $C$ is

Compute the sum of all the roots of $(2x+3)(x-4)$ $+(2x+3)(x-6)=0.$

Jamal wants to store $30$ computer files on floppy disks, each of which has a capacity of $1.44$ megabytes (mb). Three of his files require $0.8$ mb of memory each, $12$ more require $0.7$ mb each, and the remaining $15$ require $0.4$ mb each. No file can be split between floppy disks. What is the minimal number of floppy disks that will hold all the files?

Mr. Earl E. Bird leaves his house for work at exactly $8{:}00$ A.M. every morning. When he averages $40$ miles per hour, he arrives at his workplace three minutes late. When he averages $60$ miles per hour, he arrives three minutes early. At what average speed, in miles per hour, should Mr. Bird drive to arrive at his workplace precisely on time?

The sides of a triangle have lengths of $15,$ $20,$ and $25.$ Find the length of the shortest altitude.

Both roots of the quadratic equation $x^2-63x+k=0$ are prime numbers. The number of possible values of $k$ is

The digits $1,$ $2,$ $3,$ $4,$ $5,$ $6,$ $7,$ and $9$ are used to form four two-digit prime numbers, with each digit used exactly once. What is the sum of these four primes?

If $a+1$ $=b+2$ $=c+3$ $=d+4$ $=a+b+c+d+5,$ then $a+b+c+d$ is

Sarah pours four ounces of coffee into an eight-ounce cup and four ounces of cream into a second cup of the same size. She then transfers half the coffee from the first cup to the second and, after stirring thoroughly, transfers half the liquid in the second cup back to the first. What fraction of the liquid in the first cup is now cream?

A $3\times 3\times 3$ cube is formed by gluing together $27$ standard cubical dice. (On a standard die, the sum of the numbers on any pair of opposite faces is $7.$) The smallest possible sum of all the numbers showing on the surface of the $3\times 3\times 3$ cube is

Spot’s doghouse has a regular hexagonal base that measures one yard on each side. He is tethered to a vertex with a two-yard rope. What is the area, in square yards, of the region outside the doghouse that Spot can reach?

Points $A,$ $B,$ $C,$ $D,$ $E,$ and $F$ lie, in that order, on $\overline{AF},$ dividing it into five segments, each of length $1.$ Point $G$ is not on line $AF.$ Point $H$ lies on $\overline{GD},$ and point $J$ lies on $\overline{GF}.$ The line segments $\overline{HC},$ $\overline{JE},$ and $\overline{AG}$ are parallel. Find $\frac{HC}{JE}.$